#+TITLE: Stewart Platform with Flexible Elements
:DRAWER:
#+STARTUP: overview

#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

#+HTML_LINK_HOME: ./index.html
#+HTML_LINK_UP: ./index.html

#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>

#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :noweb yes
#+PROPERTY: header-args:matlab+ :mkdirp yes
#+PROPERTY: header-args:matlab+ :output-dir figs

#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:

* Simscape Model
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :results none :exports none
  simulinkproject('../');
#+end_src

** Flexible APA
#+begin_src matlab
  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(apa.n_i); length(apa.int_i); size(apa.M,1) - 6*length(apa.int_i); size(apa.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |   7 |
| Number of interface Nodes |   7 |
| Number of Modes           | 120 |
| Size of M and K matrices  | 162 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(apa.int_i)]', apa.int_i, apa.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number |   x [m] | y [m] |  z [m] |
|--------+-------------+---------+-------+--------|
|    1.0 |    697783.0 |     0.0 |   0.0 | -0.015 |
|    2.0 |    697784.0 |     0.0 |   0.0 |  0.015 |
|    3.0 |    697785.0 | -0.0325 |   0.0 |    0.0 |
|    4.0 |    697786.0 | -0.0125 |   0.0 |    0.0 |
|    5.0 |    697787.0 | -0.0075 |   0.0 |    0.0 |
|    6.0 |    697788.0 |  0.0125 |   0.0 |    0.0 |
|    7.0 |    697789.0 |  0.0325 |   0.0 |    0.0 |

** Flexible Joint
#+begin_src matlab
  flex_joint = load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(flex_joint.n_i); length(flex_joint.int_i); size(flex_joint.M,1) - 6*length(flex_joint.int_i); size(flex_joint.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |  2 |
| Number of interface Nodes |  2 |
| Number of Modes           |  6 |
| Size of M and K matrices  | 18 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(flex_joint.int_i)]', flex_joint.int_i, flex_joint.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+-------+-------+-------|
|    1.0 |    528875.0 |   0.0 |   0.0 |   0.0 |
|    2.0 |    528876.0 |   0.0 |   0.0 |  -0.0 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([1e-6*flex_joint.K(3,3), flex_joint.K(4,4), flex_joint.K(5,5), flex_joint.K(6,6)]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %0.f ');
#+end_src

#+RESULTS:
| *Caracteristic*            | *Value* |
|----------------------------+---------|
| Axial Stiffness [N/um]     |      94 |
| Bending Stiffness [Nm/rad] |       5 |
| Bending Stiffness [Nm/rad] |       5 |
| Torsion Stiffness [Nm/rad] |     260 |

** Identification
And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_platform_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
#+end_src

#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');
#+end_src

#+begin_src matlab
  disturbances = initializeDisturbances();
#+end_src

** No Flexible Elements
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
                                             'type_F', 'universal_3dof', ...
                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);
#+end_src

#+begin_src matlab
  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

** Flexible joints

#+name: fig:simscape_flex_joints
#+caption: Figure caption
[[file:figs/simscape_flex_joints.png]]

#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  %% Run the linearization
  Gj = linearize(mdl, io, options);
  Gj.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

** Flexible APA

#+name: fig:simscape_flex_apa
#+caption: Figure caption
[[file:figs/simscape_flex_apa.png]]

#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
                                             'type_F', 'universal_3dof', ...
                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  %% Run the linearization
  Ga = -linearize(mdl, io, options);
  Ga.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Ga.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

** Flexible Joints and APA

#+name: fig:simscape_flexible
#+caption: Figure caption
[[file:figs/simscape_flexible.png]]

#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  Gf = -linearize(mdl, io, options);
  Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

** Save
#+begin_src matlab
  save('./mat/flexible_stewart_platform.mat', 'G', 'Gj', 'Ga', 'Gf');
#+end_src

** Direct Velocity Feedback
#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  h = zeros(4, 1);
  h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
  h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
  h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
  h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
  legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/flexible_elements_effect_dvf.pdf', 'width', 'full', 'height', 'full');
#+end_src

#+name: fig:flexible_elements_effect_dvf
#+caption: Change of the DVF plant dynamics with the added flexible elements
#+RESULTS:
[[file:figs/flexible_elements_effect_dvf.png]]

** Integral Force Feedback
#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
  end
  h = zeros(4, 1);
  h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
  h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
  h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
  h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
  legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/flexible_elements_effect_iff.pdf', 'width', 'full', 'height', 'full');
#+end_src

#+name: fig:flexible_elements_effect_iff
#+caption: Change of the IFF plant dynamics with the added flexible elements
#+RESULTS:
[[file:figs/flexible_elements_effect_iff.png]]

** Procedure to include flexible elements into Simscape
In order to model a flexible element with only few mass-spring-damper elements:
- FEM of the flexible element
- Super-Element extraction
- Parameters to extract
  - For the flexible joint: axial, shear, bending and torsion stiffnesses
  - For the APA: k1, ka, ke, c1
- This can be done directly from the Stiffness matrix or using identification from a simple Simscape model

** Conclusion
#+begin_important
  The results seems to indicate that the model is well represented with only few degrees of freedom.
  This permit to have a relatively sane number of states for the model.
#+end_important

* Control with flexible elements
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :results none :exports none
  simulinkproject('../');
#+end_src

** Flexible APA and Flexible Joint
#+begin_src matlab
  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
  flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');
#+end_src

#+begin_src matlab
  disturbances = initializeDisturbances();
  references = initializeReferences(stewart);
#+end_src

** Identification
And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_platform_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
#+end_src

#+begin_src matlab
  G = -linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
#+end_src

** Decentralized Direct Velocity Feedback
#+begin_src matlab :exports none
  freqs = logspace(1, 4, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(G(i, i), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Diagonal elements of the Plant');

  ax2 = subplot(2, 2, 3);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$\\epsilon_{\\mathcal{L}_%i}/\\tau_%i$', i, i));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northwest');

  ax3 = subplot(2, 2, 2);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Off-Diagonal elements of the Plant');

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

Controller Design
#+begin_src matlab
  Kl = -1e5*s/(1 + s/2/pi/2e2)/(1 + s/2/pi/5e2) * eye(6);
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(1, 4, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G(1, 1)*Kl(1,1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1)*Kl(1,1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :results replace value
  isstable(feedback(G(1:6,1:6)*Kl, eye(6), -1))
#+end_src

#+RESULTS:
: 1

** HAC
#+begin_src matlab
  Kx = tf(zeros(6));

  controller = initializeController('type', 'ref-track-hac-dvf');
#+end_src

#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'stewart_platform_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],              1, 'input');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Relative Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [m]

  %% Run the linearization
  G = linearize(mdl, io);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
#+end_src

#+begin_src matlab
  Gl = -stewart.kinematics.J*G;
  Gl.OutputName  = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(1, 4, 1000);

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Diagonal elements of the Plant');

  ax2 = subplot(2, 2, 3);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$\\epsilon_{\\mathcal{L}_%i}/\\tau_%i$', i, i));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northwest');

  ax3 = subplot(2, 2, 2);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Off-Diagonal elements of the Plant');

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

#+begin_src matlab
  wc = 2*pi*300;
  Kl = diag(1./diag(abs(freqresp(Gl, wc)))) * wc/s * 1/(1 + s/3/wc);
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(1, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Kl(i, i)*Gl(i, i), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*Gl(i, i), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

* Flexible Joint Specifications
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :results none :exports none
  simulinkproject('../');
#+end_src

** Stewart Platform Initialization
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 43e6*ones(6,1), 'K1', 0.4e6*ones(6,1), 'C1', 10*ones(6,1));
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart);
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);
#+end_src

#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');
#+end_src

#+begin_src matlab
  disturbances = initializeDisturbances();
#+end_src

#+begin_src matlab
  open('stewart_platform_model.slx')
#+end_src

** Initialization                                                  :noexport:
#+begin_src matlab
  freqs = logspace(0, 3, 1000);

  c1 = [     0 0.4470 0.7410 0.2]; % Blue
  c2 = [0.8500 0.3250 0.0980 0.2]; % Orange
  c3 = [0.9290 0.6940 0.1250 0.2]; % Yellow
  c4 = [0.4940 0.1840 0.5560 0.2]; % Purple
  c5 = [0.4660 0.6740 0.1880 0.2]; % Green
  c6 = [0.3010 0.7450 0.9330 0.2]; % Light Blue
  c7 = [0.6350 0.0780 0.1840 0.2]; % Red
  colors = [c1; c2; c3; c4; c5; c6; c7];
#+end_src

#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_platform_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],              1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm');  io_i = io_i + 1; % Relative Motion - Legs [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
#+end_src

** Effect of the Bending Stiffness
#+begin_src matlab
  Kfs = [1, 10, 100, 1000]; % [Nm/rad]
#+end_src

#+begin_src matlab :exports none
  Gs = {zeros(length(Kfs))};

  for Kfs_i = 1:length(Kfs)
      stewart = initializeJointDynamics(stewart, 'type_F', 'universal', ...
                                                 'type_M', 'spherical', ...
                                                 'Kf_M', Kfs(Kfs_i)*ones(6,1), ...
                                                 'Kt_M', 0*ones(6,1), ...
                                                 'Kf_F', Kfs(Kfs_i)*ones(6,1), ...
                                                 'Kt_F', 0*ones(6,1), ...
                                                 'Ka_M', 0*ones(6,1), ...
                                                 'Kr_M', 0*ones(6,1), ...
                                                 'Ka_F', 0*ones(6,1), ...
                                                 'Kr_F', 0*ones(6,1));

      G = linearize(mdl, io, options);
      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
      Gs(Kfs_i) = {G};
  end
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Kfs_i = 1:length(Kfs)
      plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kfs(Kfs_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Kfs_i = 1:length(Kfs)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kfs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Kfs_i = 1:length(Kfs)
      plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kfs(Kfs_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Kfs_i = 1:length(Kfs)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kfs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

** Effect of the Torsion Stiffness
#+begin_src matlab
  Kts = [10, 100, 500, 1000]; % [Nm/rad]
#+end_src

#+begin_src matlab :exports none
  Gt = {zeros(length(Kts))};

  for Kts_i = 1:length(Kts)
      stewart = initializeJointDynamics(stewart, 'type_F', 'universal', ...
                                                 'type_M', 'spherical', ...
                                                 'Kf_M', 0*ones(6,1), ...
                                                 'Kt_M', Kts(Kts_i)*ones(6,1), ...
                                                 'Kf_F', 0*ones(6,1), ...
                                                 'Kt_F', Kts(Kts_i)*ones(6,1), ...
                                                 'Ka_M', 0*ones(6,1), ...
                                                 'Kr_M', 0*ones(6,1), ...
                                                 'Ka_F', 0*ones(6,1), ...
                                                 'Kr_F', 0*ones(6,1));

      G = linearize(mdl, io, options);
      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

      Gt(Kts_i) = {G};
  end
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Kts_i = 1:length(Kts)
      plot(freqs, abs(squeeze(freqresp(Gt{Kts_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'DisplayName', sprintf('$k_t = %.0f [Nm/rad]$', Kts(Kts_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Gt{Kts_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Kts_i = 1:length(Kts)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Gt{Kts_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Kts_i = 1:length(Kts)
      plot(freqs, abs(squeeze(freqresp(Gs{Kts_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kts(Kts_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Gs{Kts_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Kts_i = 1:length(Kts)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kts_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

** Effect of the Axial Stiffness
#+begin_src matlab
  Kas = [1e6, 1e7, 1e8, 5e8, 1e9]; % [N/m]
#+end_src

#+begin_src matlab :exports none
  Ga = {zeros(length(Kas))};

  for Kas_i = 1:length(Kas)
      stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
                                                 'type_M', 'spherical_3dof', ...
                                                 'Kf_M', 0*ones(6,1), ...
                                                 'Kt_M', 0*ones(6,1), ...
                                                 'Kf_F', 0*ones(6,1), ...
                                                 'Kt_F', 0*ones(6,1), ...
                                                 'Ka_M', Kas(Kas_i)*ones(6,1), ...
                                                 'Kr_M', 0*ones(6,1), ...
                                                 'Ka_F', Kas(Kas_i)*ones(6,1), ...
                                                 'Kr_F', 0*ones(6,1));

      G = linearize(mdl, io, options);
      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

      Ga(Kas_i) = {G};
  end
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Kas_i = 1:length(Kas)
      plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'DisplayName', sprintf('$k_a = %.0e [N/m]$', Kas(Kas_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Kas_i = 1:length(Kas)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Ga{Kas_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Kas_i = 1:length(Kas)
      plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'DisplayName', sprintf('$k_a = %.0e [N/m]$', Kas(Kas_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Kas_i = 1:length(Kas)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Ga{Kas_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

** Effect of the Radial (Shear) Stiffness
#+begin_src matlab
  Krs = [1e4, 1e5, 1e6, 1e7]; % [N/m]
#+end_src

#+begin_src matlab :exports none
  Gr = {zeros(length(Kas))};

  for Krs_i = 1:length(Krs)
      stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
                                                 'type_M', 'spherical_3dof', ...
                                                 'Kf_M', 0*ones(6,1), ...
                                                 'Kt_M', 0*ones(6,1), ...
                                                 'Kf_F', 0*ones(6,1), ...
                                                 'Kt_F', 0*ones(6,1), ...
                                                 'Ka_M', 1e15*ones(6,1), ...
                                                 'Kr_M', Krs(Krs_i)*ones(6,1), ...
                                                 'Ka_F', 1e15*ones(6,1), ...
                                                 'Kr_F', Krs(Krs_i)*ones(6,1));

      G = linearize(mdl, io, options);
      G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
      G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                      'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

      Gr(Krs_i) = {G};
  end
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Krs_i = 1:length(Krs)
      plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'DisplayName', sprintf('$k_r = %.0e [N/m]$', Krs(Krs_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Krs_i = 1:length(Krs)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Gr{Krs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;

  for Krs_i = 1:length(Krs)
      plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'DisplayName', sprintf('$k_r = %.0e [N/m]$', Krs(Krs_i)));
      for i = 2:6
          plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'HandleVisibility', 'off');
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for Krs_i = 1:length(Krs)
      for i = 1:6
          plot(freqs, 180/pi*angle(squeeze(freqresp(Gr{Krs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :));
      end
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

** Comparison of perfect joint and worst specified joint
#+begin_src matlab :exports none
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', ...
                                             'type_M', 'spherical_p');

  Gp = linearize(mdl, io, options);
  Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gp.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
                                             'type_M', 'spherical_3dof', ...
                                             'Kf_M', 100*ones(6,1), ...
                                             'Kt_M', 500*ones(6,1), ...
                                             'Kf_F', 100*ones(6,1), ...
                                             'Kt_F', 500*ones(6,1), ...
                                             'Ka_M', 200e6*ones(6,1), ...
                                             'Kr_M', 1e6*ones(6,1), ...
                                             'Ka_F', 200e6*ones(6,1), ...
                                             'Kr_F', 1e6*ones(6,1));

  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
  end
  plot(freqs, abs(squeeze(freqresp(Gp('L1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('Fm1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
  end
  plot(freqs, abs(squeeze(freqresp(Gp('Fm1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

** Conclusion
Qualitatively:
|                   | *Specification*                                  |
|-------------------+--------------------------------------------------|
| Axial Stiffness   | Much larger than the Actuator axial stiffness    |
| Shear Stiffness   |                                                  |
| Bending Stiffness | Much smaller than the Actuator bending stiffness |
| Torsion Stiffness |                                                  |

Quantitatively:
|                   | *Specification* |
|-------------------+-----------------|
| Axial Stiffness   | > 200  [N/um]   |
| Shear Stiffness   | > 1 [N/um]      |
| Bending Stiffness | < 100  [Nm/rad] |
| Torsion Stiffness | < 500  [Nm/rad] |

* Flexible Joint Specifications with the APA300ML
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :results none :exports none
  simulinkproject('../');
#+end_src

** Stewart Platform Initialization
#+begin_src matlab
  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
  stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
  stewart = initializeCylindricalPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);
#+end_src

#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');
#+end_src

#+begin_src matlab
  disturbances = initializeDisturbances();
#+end_src

#+begin_src matlab
  open('stewart_platform_model.slx')
#+end_src

** Initialization                                                  :noexport:
#+begin_src matlab
  freqs = logspace(0, 3, 1000);

  c1 = [     0 0.4470 0.7410 0.2]; % Blue
  c2 = [0.8500 0.3250 0.0980 0.2]; % Orange
  c3 = [0.9290 0.6940 0.1250 0.2]; % Yellow
  c4 = [0.4940 0.1840 0.5560 0.2]; % Purple
  c5 = [0.4660 0.6740 0.1880 0.2]; % Green
  c6 = [0.3010 0.7450 0.9330 0.2]; % Light Blue
  c7 = [0.6350 0.0780 0.1840 0.2]; % Red
  colors = [c1; c2; c3; c4; c5; c6; c7];
#+end_src

#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_platform_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],              1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm');  io_i = io_i + 1; % Relative Motion - Legs [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
#+end_src

** Comparison of perfect joint and worst specified joint
#+begin_src matlab :exports none
  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', ...
                                             'type_M', 'spherical_p');

  Gp = linearize(mdl, io, options);
  Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  Gp.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};

  stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
                                             'type_M', 'spherical_3dof', ...
                                             'Kf_M', 100*ones(6,1), ...
                                             'Kt_M', 500*ones(6,1), ...
                                             'Kf_F', 100*ones(6,1), ...
                                             'Kt_F', 500*ones(6,1), ...
                                             'Ka_M', 200e6*ones(6,1), ...
                                             'Kr_M', 1e6*ones(6,1), ...
                                             'Ka_F', 200e6*ones(6,1), ...
                                             'Kr_F', 1e6*ones(6,1));

  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                  'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
  end
  plot(freqs, abs(squeeze(freqresp(Gp('L1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :exports none
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('Fm1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
  end
  plot(freqs, abs(squeeze(freqresp(Gp('Fm1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

* Relative Motion Sensors
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :results none :exports none
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open('stewart_platform_model.slx')
#+end_src

** Stewart Platform Initialization
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart);
  stewart = generateGeneralConfiguration(stewart);
  stewart = computeJointsPose(stewart);
#+end_src

#+begin_src matlab
  apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
  stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
  % stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
#+end_src

#+begin_src matlab
  flex_joint = load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
  stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
                                             'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
                                             'type_F', 'universal_3dof', ...
                                             'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
                                             'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
                                             'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
                                             'Kt_F', flex_joint.K(6,6)*ones(6,1));
#+end_src

#+begin_src matlab
  stewart = initializeCylindricalPlatforms(stewart);
#+end_src

#+begin_src matlab
  stewart = initializeCylindricalStruts(stewart);
#+end_src

#+begin_src matlab
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);

  references = initializeReferences(stewart);
#+end_src

#+begin_src matlab
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 50);
  controller = initializeController('type', 'open-loop');
#+end_src

#+begin_src matlab
  disturbances = initializeDisturbances();
#+end_src

** Initialization                                                  :noexport:
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_platform_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');              io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm');  io_i = io_i + 1; % Relative Motion - Actuators [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Lm');   io_i = io_i + 1;  % Relative Motion - Legs [m]
#+end_src

#+begin_src matlab :exports none
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
                  'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(1, 4, 5000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Actuator Stroke');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
  end
  plot(freqs, abs(squeeze(freqresp(G('D1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Leg Stroke');
  for i = 2:6
      plot(freqs, abs(squeeze(freqresp(G(sprintf('D%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'northeast');

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('D%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
  exportFig('figs/comp_relative_motion_sensor_act_leg.pdf', 'width', 'full', 'height', 'full');
#+end_src

#+name: fig:comp_relative_motion_sensor_act_leg
#+caption: Comparison of the dynamique from actuator to relative motion sensor located in parallel with the actuator or with the leg (flexible joints included)
#+RESULTS:
[[file:figs/comp_relative_motion_sensor_act_leg.png]]

* Struts with Encoders
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :results none :exports none
  simulinkproject('../');
#+end_src

** Flexible Strut
#+begin_src matlab
  apa = load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
  data2orgtable([length(apa.n_i); length(apa.int_i); size(apa.M,1) - 6*length(apa.int_i); size(apa.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
#+end_src

#+RESULTS:
| Total number of Nodes     |  8 |
| Number of interface Nodes |  8 |
| Number of Modes           |  6 |
| Size of M and K matrices  | 54 |

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([[1:length(apa.int_i)]', apa.int_i, apa.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
#+end_src

#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number |   x [m] |  y [m] |    z [m] |
|--------+-------------+---------+--------+----------|
|    1.0 |    504411.0 |     0.0 |    0.0 |   0.0405 |
|    2.0 |    504412.0 |     0.0 |    0.0 |  -0.0405 |
|    3.0 |    504413.0 | -0.0325 |    0.0 |      0.0 |
|    4.0 |    504414.0 | -0.0125 |    0.0 |      0.0 |
|    5.0 |    504415.0 | -0.0075 |    0.0 |      0.0 |
|    6.0 |    504416.0 |  0.0325 |    0.0 |      0.0 |
|    7.0 |    504417.0 |   0.004 | 0.0145 | -0.00175 |
|    8.0 |    504418.0 |   0.004 | 0.0166 | -0.00175 |

** Stewart Platform
#+begin_src matlab
  stewart = initializeStewartPlatform();
  stewart = initializeFramesPositions(stewart, 'H', 95e-3, 'MO_B', 220e-3);
  stewart = generateGeneralConfiguration(stewart, 'FH', 22.5e-3, 'FR', 114e-3, 'FTh', [   -11,    11, 120-11, 120+11, 240-11, 240+11]*(pi/180), ...
                                                  'MH', 22.5e-3, 'MR', 110e-3, 'MTh', [-60+15, 60-15,  60+15, 180-15, 180+15, -60-15]*(pi/180));
  stewart = computeJointsPose(stewart);

  stewart = initializeFlexibleStrutAndJointDynamics(stewart, 'H', (apa.int_xyz(1,3) - apa.int_xyz(2,3)), ...
                                                             'K', apa.K, ...
                                                             'M', apa.M, ...
                                                             'n_xyz', apa.n_xyz, ...
                                                             'xi', 0.1, ...
                                                             'Gf', -2.65e7, ...
                                                             'step_file', 'mat/APA300ML.STEP');

  stewart = initializeSolidPlatforms(stewart);
  stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
  stewart = computeJacobian(stewart);
  stewart = initializeStewartPose(stewart);
  stewart = initializeInertialSensor(stewart);
#+end_src

#+begin_src matlab
  disturbances = initializeDisturbances();
  ground = initializeGround('type', 'none');
  payload = initializePayload('type', 'rigid', 'm', 1);
  controller = initializeController('type', 'open-loop');
  references = initializeReferences(stewart);
#+end_src


#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'stewart_platform_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
  io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Lm'); io_i = io_i + 1; % Force Sensors [N]
#+end_src

#+begin_src matlab
  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
  G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ...
                  'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
  end
  for i = 1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

  linkaxes([ax1,ax2],'x');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_relative_motion_sensor_act_leg_encoder.pdf', 'width', 'full', 'height', 'full');
#+end_src

#+name: fig:comp_relative_motion_sensor_act_leg_encoder
#+caption:
#+RESULTS:
[[file:figs/comp_relative_motion_sensor_act_leg_encoder.png]]